Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000675
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[2,1,4,3] => [2,2]
 => [2]
 => [1,0,1,0]
 => 1
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[3,4,1,2] => [2,2]
 => [2]
 => [1,0,1,0]
 => 1
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[4,3,2,1] => [2,2]
 => [2]
 => [1,0,1,0]
 => 1
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,1,4,5,3] => [3,2]
 => [2]
 => [1,0,1,0]
 => 1
[2,1,5,3,4] => [3,2]
 => [2]
 => [1,0,1,0]
 => 1
[2,1,5,4,3] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3,1,4,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[2,3,1,5,4] => [3,2]
 => [2]
 => [1,0,1,0]
 => 1
[2,4,3,1,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[2,4,5,1,3] => [3,2]
 => [2]
 => [1,0,1,0]
 => 1
[2,5,3,4,1] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[2,5,4,3,1] => [3,2]
 => [2]
 => [1,0,1,0]
 => 1
[3,1,2,4,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[3,1,2,5,4] => [3,2]
 => [2]
 => [1,0,1,0]
 => 1
[3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[3,2,1,5,4] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,2,4,1,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[3,2,5,4,1] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[3,4,1,2,5] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,4,1,5,2] => [3,2]
 => [2]
 => [1,0,1,0]
 => 1
[3,4,5,2,1] => [3,2]
 => [2]
 => [1,0,1,0]
 => 1
Description
The number of centered multitunnels of a Dyck path. This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.